# std::sph_legendre, std::sph_legendref, std::sph_legendrel

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< cpp‎ | numeric‎ | special functions

Defined in header `<cmath>` |  |
| (1) |
float sph_legendre ( unsigned l, unsigned m, float theta ); double sph_legendre ( unsigned l, unsigned m, double theta ); long double sph_legendre ( unsigned l, unsigned m, long double theta ); |  | (since C++17) (until C++23)
/* floating-point-type */ sph_legendre( unsigned l, unsigned m, /* floating-point-type */ theta ); |  | (since C++23)
float sph_legendref( unsigned l, unsigned m, float theta ); | (2) | (since C++17)
long double sph_legendrel( unsigned l, unsigned m, long double theta ); | (3) | (since C++17)
Additional overloads |  |
Defined in header `<cmath>` |  |
template< class Integer > double sph_legendre ( unsigned l, unsigned m, Integer theta ); | (A) | (since C++17)

1-3) Computes the spherical associated Legendre function of degree l, order m, and polar angle theta. The library provides overloads of `std::sph_legendre` for all cv-unqualified floating-point types as the type of the parameter theta.(since C++23)
A) Additional overloads are provided for all integer types, which are treated as double.

### Parameters

l | - | degree
m | - | order
theta | - | polar angle, measured in radians

### Return value

If no errors occur, returns the value of the spherical associated Legendre function (that is, spherical harmonic with ϕ = 0) of l, m, and theta, where the spherical harmonic function is defined as Ym
l(theta,ϕ) = (-1)m
[

(2l+1)(l-m)!
4π(l+m)!

]1/2
Pm
l(cos(theta))eimϕ
where Pm
l(x) is std::assoc_legendre(l, m, x)) and |m|≤l.

Note that the Condon-Shortley phase term (-1)m
is included in this definition because it is omitted from the definition of Pm
l in std::assoc_legendre.

### Error handling

Errors may be reported as specified in math_errhandling.

- If the argument is NaN, NaN is returned and domain error is not reported.
- If l≥128, the behavior is implementation-defined.

### Notes

Implementations that do not support C++17, but support ISO 29124:2010, provide this function if `__STDCPP_MATH_SPEC_FUNCS__` is defined by the implementation to a value at least 201003L and if the user defines `__STDCPP_WANT_MATH_SPEC_FUNCS__` before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header `tr1/cmath` and namespace `std::tr1`.

An implementation of the spherical harmonic function is available in boost.math, and it reduces to this function when called with the parameter phi set to zero.

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their argument num of integer type, std::sph_legendre(int_num1, int_num2, num) has the same effect as std::sph_legendre(int_num1, int_num2, static_cast<double>(num)).

### Example

Run this code

    #include <cmath>
    #include <iostream>
    #include <numbers>

    int main()
    {
        // spot check for l=3, m=0
        double x = 1.2345;
        std::cout << "Y_3^0(" << x << ") = " << std::sph_legendre(3, 0, x) << '\n';

        // exact solution
        std::cout << "exact solution = "
                  << 0.25 * std::sqrt(7 / std::numbers::pi)
                      * (5 * std::pow(std::cos(x), 3) - 3 * std::cos(x))
                  << '\n';
    }

Output:

    Y_3^0(1.2345) = -0.302387
    exact solution = -0.302387

### See also

assoc_legendreassoc_legendrefassoc_legendrel (C++17)(C++17)(C++17) | associated Legendre polynomials (function)

### External links

Weisstein, Eric W. "Spherical Harmonic." From MathWorld — A Wolfram Web Resource.

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